Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Abstract

The relative entropy of entanglement ER is defined as the distance of a multi-partite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a (unique) closest separable state, even in infinite dimension; also, ER is everywhere lower semi-continuous. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multi-partite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity -more generally, all relative entropy distances from the sets of states with non-negative λ-quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states. We complement our findings by giving explicit and asymptotically tight continuity estimates for ER and closely related quantities in the presence of an energy constraint. CONTENTS I. Introduction II. Notation A. Topologies for quantum systems B. Relative entropy of resource III. Main results A. The statements B. Proofs C. On the continuity of the relative entropy of resource D. On the continuity of the minimiser IV. Applications A. Relative entropy of entanglement B. Relative entropy of multi-partite entanglement C. Relative entropy of NPT entanglement D. Relative entropy of non-classicality, Wigner non-positivity, and generalisations thereof E. Relative entropy of non-Gaussianity V. Tight uniform continuity bounds for the relative entropy of entanglement and generalisations thereof A. Energy constraints B. Uniform continuity on energy-constrained states: bipartite case C. Uniform continuity on energy-constrained states: multipartite case VI. Conclusions and outlook VII. Acknowledgements References A. On the proof of an inequality involving the multi-partite relative entropy of entanglement

Continuity of characteristics of composite quantum systems: a review

“…Theorem 7 with the function G = G ℓ,ω can be formulated as Corollary 5. [18] Let H A 1 be the operator defined in (54).…”

“…If H A 1 is the grounded Hamiltonian of the ℓ-mode quantum oscillator with the frequencies ω 1 , ..., ω ℓ defined in (54) then the function F H A 1 satisfies condition (15) and the role of function G in Theorem 8 can be played by the function G ℓ,ω defined in (55) that satisfies conditions ( 49),( 50) and (51). In this case continuity bound ( 57) is asymptotically tight for large E for any function f satisfying condition (58).…”

“…• the relative entropy of entanglement, the relative entropy distance to the PPT states and the Rains bound in bipartite quantum systems [104,105,106,107], all these characteristics belong to the classes L 1 2 (1, 1) and L 2 2 (1/2, 1) [54];…”

Computable lower bounds on the entanglement cost of quantum channels

“…For example, picking an orthonormal basis {| } ∈N of H, one sees that the sequence (| |) →N has no limit with respect to the former topology, yet it satisfies | | * − −− → →∞ 0. A word of caution before proceeding is advisable: the weak* topology considered here is not the same commonly used in the von Neumann algebra approach to quantum theory [42,Remark 2].…”

“…Remark. The use of the weak* topology in the context of quantum resource theories of states was explored in [38,39,42,46]. Building on that, the possibility of extending these concepts to spaces of quantum channels was considered in [47].…”